Proposition 9

Let M be an A-module and $(P, u)$ a projective cover of M. Denote the radical of the ring A by $\mathfrak{r}$. The homomorphism $\overline{u}: P/\mathfrak{r}P\rightarrow M/\mathfrak{r}M$ deduced from $u$ by passing to the quotients is an isomorphism.

The homomorphism $u$ is surjective by definition, so $\overline{u}$ is surjective. Denote the kernel of $u$ by N. We have $u^{-1}(\mathfrak{r}M) = N +\mathfrak{r}P$. Let us prove that we have $N\subset \mathfrak{r}P$, which implies the injectivity of $\overline{u}$. For every maximal submodule $P'$ of P, we have $u(P')\not= M$, hence $P'+ N\not= P$; since $P'$ is maximal, we have $N\subset P'$. The submodule N of P is therefore contained in the radical of P. Now, the latter is equal to $\mathfrak{r}P$ by Proposition 6 of VIII, p. 158.